FUNCTION. 
=VdV, whence F pene given a function of the 
pace ve may be deduced a funétio 
de d’ x 
i a 24 &c. if the 
Since A x a3 At+— dn . (S87 + &e. 
. d 
body fall from reft, or the velocity ic =o, then A x 
a’ x dix 
= ~—— . (Ar &c. now 
1.2.daf° a erent: oe 
fhall be 
ad}: 
1.2.3.d# 
. than the fum of all fucceeding terms, fo that 
+ = At &e.) 4 (> ae 
ao I.2. oe cl 2s “TF 
fince A ¢ may be taken fuch, that 
+ aoa. «A t) and & fortiori, lefs, for all leffer 
feds 300? 
values of A #, it is clear, that be taking Af of a proper 
a At) (A #)? 
. (A2)* by a dif- 
da 
{mallnefs, Ge de + ——. eo + 
may be made to differ from : ee . 
ference lefs than any affignable quantity. 
‘This is all that is to be underflood in the prefent method, 
concerning an equality that is faid to fubfilt between the 
{paces defcribed in initio motus, the fquares of the times, 
and o ther conftant peti ; 
As, cafe of tangents and radii of curvature, the 
analytical properties were firft tated and the geometrical de- 
oe d, fo, in the c of velocities a accelerating forces, 
ated, a mechanical 
noone ed ; 
Soa aay that co 
e de- 
feribed by another uniform motion, the veloc city may 
re de- 
duced; thus, let a be its meafure, then the 
feribed in thetime A # = @. As, and Ax being = aT A 
ae (At)® + &e. 
.F 1.2.02 
da 
ea oo erg ~ (Arty + &.is < 
dv v roo 
(SF —~ajyAt+t ne a Ve ~ (Az)? + &e. which 
cannot be generally afferted ; for, make the conftant quan- 
tity « =<, then A # may be taken fo fmall, that 
a x 7 
rarer (4#) + &e, hall be < (G= — =) Art 
a . (At)? + &e. but if = = a, then can 
(2) - At + &e. be <(- ars 
t 
ee ** 5 oF Ay + &e. whiltt 4 ra does not equal «, and 
ar el the element a put for the velocity, mult 
= or 
n like manner taking the definition that has been given 
of rs accelerating force, itg value may be fhewn equal 
d’ x 
1.2.d2°" 
e term venue! has caufed much confufion of idea: 
its meafure is the conftant quantity exprefling the ratio be 
tween the {pace ae time in uniform motion ; in the equa 
tion « = a#, it is a, which element ferves to cao one 
uniform motion from another ; in variable motion, the m 
fure of the velocity is ;-— ; and this only is meant, that a 
dt 
body moving uniformly with this velocity $3 would de- 
: d : a 
aa . At, which has the conditions above 
- 
varies as fe. 
7 
dw x , 
The meee ae a + &e. 
nd confe tl aed A &e. 1 
a quen ys es eae + &e. now 
~ may be made to differ from 
— 
id what has ee ee ay 
oF by a quantity lefs than any affignable quantity ; and 
this: isall that is clearly mae concerning any ee ity that 
can fubfilt between — 
dé 
* and a certain ftate oe re in the 
ee : Ax Ax 
1 limits, the limit of — —— Is fal 
anguage of limits, the limit o <7 ork i ; 38 faid to equal 
a 
a? 
The de eee of baal being giv yen, to nine its 
quantity, or, meafure in variable motion, enon m 
of aia : ana patie a function of i cae the 
we 
differential co-efficient Ty the meafure of the velacity 
ire 
muft be pega Reba? preceding sates can al- 
ential co-efficie: x what 
ey 
- the quantity deduced muft be what, really, — is; not fim. 
oe 
ply the firft differential co-efficient o x expanded, 
but the firft differential co-efficient ain ed bya a particular 
hy othefis ; now the iit aay fluxionifts have fallowed 
is this; they define flusio e velacities, a havin 
certain pr oceffes, obtained. the os fy mbol 
ocity, bee in fact, had obtained the fox on or or differe ntial 
of a quantity : a procedure, it fhould ion, not very philos 
fophiea! fince velocity, asa term by which fluxions are 
3Na defined, 
